Thursday, 16 November 2017

Answer to The Imitation Game

I have two children. One of them is a boy and they were born on a Tuesday.

What is the probability that both children are boys?

This is a hard question, and Ben ****ed up the explanation when he tried to do it live. So, as penance, we made him sit down and explain it as a video.

Here's a simpler question written out much nicer:

I have two children. One of them is a boy.
What is the probability that both children are boys?

Now you may think the probability is 50%, but that is not so (note that we are assuming that boy and girl births are equally likely). The reason is because we have more information about the children.

Suppose we denote a boy by "b" and a girl by "g". Further, we capitalise the letter to denote the elder child. In this way we could have the following combinations of children:

Bb

Gb

Bg

Gg

However, we know we have at least one boy, so we can't have Gg. Out of the possibilities that are left, namely Bb, Gb and Bg, there is only one way to get two boys, the chance is 1/3! Counter-intuitive no?

Note that if we had posed the problem as I have two children and my eldest is a boy then (using the above argument) the probability of have a second boy is then 1/2.

Probability can be a tricksy animal. Even for a Cambridge educated lecturer!

2 comments:

Since the "Tuesday" question was introduced at a conference named for Martin Gardner, and he is the man who popularized your simpler question ("Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"), we should consider his answer. Which ultimately was that there is not enough information given to answer it. He admitted that his question was ambiguous, but it was less ambiguous than yours.

The only possible answer to both of your questions is 1/2.

Probability does not ask us "what possibilities allow this information to be true," it asks "what possibilities would result in this statement being made." If the parent in your riddle has a boy and a girl, then (s)he had to choose between asking the question about boys, or about girls, and it produces a paradox if you assume it had to be "about boys" just because it was."

It's called Bertrand's Box Paradox, and the only difference when compared to yours is the number of possible cases. Add a fourth box to his, with a gold and a silver coin, and it is identical.

Most modern treatments will use that name for the problem, but Bertrand used it like this: Suppose I tell you that I have two children, and have written the gender of (at least) one on a hidden note card. What is the probability both children have that gender?

If what I wrote is "boy," this is identical to your question. If what I wrote is "girl," it is a functionally equivalent question that must have the same answer. Since the answer must be the same regardless of what I wrote, you don't need to see the note card. The answer to this question is the same as yours.

But you have no information that would allow it to be anything other than the probability that my two children have the same gender, which is 1/2.